To factor \( x^3 + x^2 - 8x - 12 \), we'll follow these steps:

### 1. Group terms:
We can attempt to factor by grouping. Group the terms in pairs:
\[
(x^3 + x^2) - (8x + 12)
\]

### 2. Factor out the greatest common factor (GCF) from each group:
\[
x^2(x + 1) - 4(2x + 3)
\]

### 3. Check for a common factor between the two groups:
The factors \( (x + 1) \) and \( (2x + 3) \) are not the same, so factoring by grouping isn't enough. 

### 4. Use the Rational Root Theorem:
Now, let's use the Rational Root Theorem to try possible rational roots. The possible rational roots are factors of the constant term (-12) divided by the factors of the leading coefficient (1). Thus, the possible rational roots are:
\[
\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12
\]

We'll test these values in the polynomial \( x^3 + x^2 - 8x - 12 \).

### 5. Test possible rational roots:
Testing \( x = 2 \):
\[
f(2) = 2^3 + 2^2 - 8(2) - 12 = 8 + 4 - 16 - 12 = -16 \quad (\text{not a root})
\]
Testing \( x = -2 \):
\[
f(-2) = (-2)^3 + (-2)^2 - 8(-2) - 12 = -8 + 4 + 16 - 12 = 0 \quad (\text{root found!})
\]

Since \( x = -2 \) is a root, \( (x + 2) \) is a factor.

### 6. Perform synthetic division:
Now, divide \( x^3 + x^2 - 8x - 12 \) by \( (x + 2) \) using synthetic or long division.

\[
\begin{array}{r|rrrr}
-2 & 1 & 1 & -8 & -12 \\
   &   & -2 & 2  & 12  \\
\hline
   & 1 & -1 & -6 & 0   \\
\end{array}
\]

The quotient is \( x^2 - x - 6 \).

### 7. Factor the quadratic:
Now, factor \( x^2 - x - 6 \). It factors as:
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

### 8. Final factorization:
Thus, the complete factorization of \( x^3 + x^2 - 8x - 12 \) is:
\[
(x + 2)(x - 3)(x + 2)
\]

This can be simplified to:
\[
(x + 2)^2(x - 3)
\]

### Final Answer:
\[
(x + 2)^2(x - 3)
\]

Do you want more details or have any questions?

Here are 5 related questions you might find useful:
1. How do you perform synthetic division for polynomials?
2. What is the Rational Root Theorem and how is it used?
3. How can you factor higher degree polynomials systematically?
4. What are other methods of factoring cubic polynomials?
5. How do you solve polynomial equations after factoring?

**Tip:** Always check for rational roots early when factoring polynomials using the Rational Root Theorem—it can save time!

This guide provides a step-by-step solution for factoring the cubic polynomial x^3 + x^2 - 8x - 12. It covers essential algebraic concepts such as polynomial factorization, the Rational Root Theorem, and synthetic division, suitable for Grades 10-12.

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